Physical systems can be modeled mathematically to simulate their behavior under different conditions. A wide variety of means exists to model physical systems, ranging from the very simplistic to the extremely complicated. One of the more complicated means to model physical systems is through the use of finite element analysis. As the name implies, finite element analysis involves a representation of individual, finite elements of a physical system in a mathematical model and the solution of this model in the presence of a predetermined set of boundary conditions.
In finite element modeling, the region that is to be analyzed is broken up into sub-regions called elements. This process of dividing the region into sub-regions may be referred to as discretization, or mesh generation. The region is represented by functions defined over each element. This generates a number of local functions that are much simpler than those which would be required to represent the entire region. The next step in finite element modeling is to analyze the response for each element. This is accomplished by building a matrix that defines the properties of the various elements within the region and a vector that defines the forces acting on each element in the region. Once all the element matrices and vectors have been created, they are combined into a structure matrix equation. This equation relates nodal responses for the entire structure to nodal forces. After applying boundary conditions, the structure matrix equation can be solved to obtain unknown nodal responses. Intra-element responses can be interpolated from nodal values using the functions that were defined over each element.
As indicated above, finite element modeling involves the creation of a mesh of finite elements. The elements are defined by nodes within the problem space. The nodes are simply points in space. The lines between the nodes are referred to as edges. The mesh is typically a structured mesh. In other words, the mesh is defined in three dimensions so that the elements within the problem space are hexahedrons. For example, they may be cubes or rectangular prisms (equivalently, in two-dimensional problems, the elements would be rectangles—see FIG. 1 for an illustration of these elements). The edges of the hexahedrons are co-incident with the edges between the nodes of the mesh. In a simple model, the nodes of the mesh may be regularly spaced to define cubic elements. It is not necessary, however, for the nodes to be evenly spaced in all finite element models. A variety of different sizes and shapes of hexahedral elements can be defined within a single mesh.
One of the problems with meshes that use hexahedron elements is that they do not lend themselves well to modeling complex geometries. In other words, it may be difficult to place nodes sufficiently close to regular surfaces, such as a well bore, to accurately determine a solution to the model along those surfaces. Viewed another way, it is difficult to fill any regularly-shaped object with regularly-shaped boxes. This situation is further complicated by limitations on the minimum distance between nodes in a structured mesh. Even when additional node elements can be defined in the mesh to improve the accuracy of the model, the inclusion of these additional elements causes the resources needed to solve the resulting model to increase. With increased accuracy, a greater number of elements is required, requiring greater computing power to generate a solution.
One way to reduce the cost associated with increasing the accuracy of a finite element model is to use an unstructured mesh, such as that disclosed in related US. Patent Application Publication No. 2002/0032550 entitled, “A METHOD FOR MODELING AN ARBITRARY WELL PATH IN A HYDROCARBON RESERVOIR USING ADAPTIVE MESHING,” filed on Jun. 29, 2001 (the “Adaptive Meshing Application”), which is hereby fully incorporated by reference. In an unstructured mesh, the elements that comprise the mesh are not constrained to fit within a predetermined structural scheme. For example, the elements are not required to be selected from a finite set of pre-defined element shapes (e.g., hexahedral) or sizes. More particularly, it may be convenient to employ a mesh that comprises simplex elements that have no pre-defined constraints. The faces of the simplex elements can be oriented to follow the contours of a surface that is being modeled and may, therefore, achieve accuracy that is on par wit a structured mesh having many more elements because the faces of the simplex elements need not be parallel to pre-determined planes. As a result, the unstructured mesh is simpler and can be solved faster and easier. The Adaptive Meshing Application discloses a method for defining the elements of an unstructured mesh such that a surface being modeled is essentially coincident with the faces of the elements in the mesh (or more particularly with some of the element faces), such that the surface can be accurately defined and irregular surfaces can be properly modeled.
However, existing meshing techniques for modeling of physical systems typically represent singularities, such as a well bore within a hydrocarbon reservoir, as a point source. Rather than modeling a well bore in detail, existing methods and systems for modeling of a reservoir system represent the well bore as a single point, a singularity, within the mesh defining a system and use a technique to disperse the singularity away so as not to have to solve the resulting model at the singularity. In essence, prior art methods disperse away this spike in the physical system and distribute it into the neighboring regions to form a homogeneous hump, which is much easier to solve mathematically than if the singularity were modeled with its own mesh. This method works well if the system to be modeled is homogenous around the well bore. Mathematically, if the system is homogenous in the area near the hump, an accurate solution can be obtained. However, the area near a well bore in a hydrocarbon reservoir is typically not homogenous. It is, however, an area of great interest within the drilling and exploration market.
Property variations in the area near the well bore greatly impact the production of a well. Further, properties such as viscosity and flow can change rapidly near the well bore. No matter how large a reservoir is (for example, a reservoir spread out over a 2-acre land area), all of the fluid extracted from the reservoir will go to the surface through a small opening (e.g., a well bore, typically, six inches to a foot in diameter). If the properties around the well bore are very poor, then it will be much more difficult to extract fluid from the reservoir to the surface. Conversely, if the properties around the well bore are good (healthy), then fluid flow to the surface will be much easier. It can be much easier to predict variations in the properties near the well bore and, hence, to predict the production capacity of a well, if the well bore is modeled using a meshing technique then it is using currently existing reservoir-only meshing modeling techniques.
Further, by being able to more accurately predict the properties near the well bore (e.g., about six inches to a few feet out from the well center line), decisions can be made on whether to make certain improvements to the well to improve flow. For example, factors such as erosion around the well bore, and changes to other properties that can affect fluid flow at the well bore, can be compensated for. Erosion and other damage to the well can also be minimized by predicting the effect on some parameter(s) of varying other parameters, such as porosity or fluid flow velocity. Prior art methods do not provide production engineers this ability to accurately determine the conditions at critical points near the well bore. As a result, production from a well may not be optimized and, in fact, well damage may occur more quickly, resulting in a loss of time and money to either abandon a well or to redrill a well.
Prior art methods and systems for modeling reservoirs also do not provide the ability to alter the values of properties in the near well region to predict changes that may occur as a result of such a change. Because existing methods disperse the well bore singularity away and do not provide solutions for within the well bore, property values in the near well bore region cannot be altered to help in predicting the effects of changing well and reservoir parameters, such as flow and extraction rate. Production engineers cannot assign values to properties within the well bore because prior art methods do not provide a mesh having great enough resolution in the near well bore region to allow modeling and alteration of property values within the well bore region.